
Since the function \((\varphi -1)/L(\varphi )\)  is a contraction and \(\varphi (0)-1=0\) , the Rademacher comparison inequality [1]} implies
\(\mathbb {E}_{\varepsilon }\bigg \Vert n^{-1}\sum _{i=1}^n \varepsilon _i \delta _{X_i}\bigg \Vert _{\mathcal {G}_{\varphi }} &\le 2L(\varphi )\mathbb {E}_{\varepsilon }\bigg \Vert n^{-1}\sum _{i=1}^n \varepsilon _i \delta _{X_i}\bigg \Vert _{\mathcal {F}} \\&=2L(\varphi ) R_n(\mathcal {F}) .\) 
